The Prime Ideal Structure of the Minkowski Ring of Polytopes

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Journal of Pure and Applied Algebra 7X (1992) 239-251 239 North-Holland

The prime ideal structure of the Minkowski ring of polytopes

Klaus G. Fischer and Jay Shapiro Department of Mathematical Sciences. George Mason lJniver.sit~, Fairfax, VA 220_30. USA

Communicated by C.A. Weihel Received 25 March 1YYl Revised 28 May 1991

Ahstracr

Fischer, K.G. and J. Shapiro, The prime ideal structure of the Minkowski ring of polytopes, Journal of Pure and Applied Algebra 78 (1992) 239-251.

We examine the suhrings of the Minkowski ring which are generated hy finitely many polytopes. Such a ring is the homomorphic image of the ring of polynomials in a finite number of variables with cocfficicnts in Z. We relate the prime ideal structure of this ring to the rational relations on the vertices of the polytopes.

Introduction

The convex hull of a finite set of points in the real d-dimensional Euclidean space KY’is called a polytope. The additive group of functions generated by the characteristic functions of all the polytopes in R” is called the group of simple functions on the polytopes which we denote by M”. If P and Q are polytopes and c is the characteristic function, then one may define a multiplication on M” given by c(P). c(Q) = c(P + Q), where P+Q={p+q: pEP, qEQ} is again a polytope called rhe Minkowski sum of P and Q. It is known [3,6] that this multiplication defines a commutative ring structure on M” and using the terminol- ogy established by Lawrence in [6], we call this the Minkowski ring of polytopes in ET’. The identity element of M” is the characteristic function of the zero-vector in R”. In the paper we will suppress notation and write f = C a,P,, where the a,‘s are integers, to mean the simple function f = c a,c(P,) and write P. Q to mean c(P+ Q>.

Corresporldence to: K.G. Fischer, Department of Mathematical Sciences. George Mason Univer- sity. Fairfax. VA 22030. USA.

Historically, the interest in the Minkowski sum was due to work by Groemer [3] to extend Minkowski’s geometric idea of mixed volumes to a wider class of convex sets. Indeed, Groemer was the first to show that M” is a commutative ring and points out that M’ has zero divisors (see Example 1.5). Lawrence in [6], focuses more sharply on the algebraic structure to easily deduce some reciprocity theorems. He goes on, as does Groemer, to show that this ring structure exists for a wider class of convex sets than just the polytopes. McMullen in [7] studies the closely related polytope algebras whose multiplication again depends on the Minkowski sum. If {P,, . ., P,} is a finite set of polytopes in R”, then the subring of Md generated by these polytopes and the identity element will be denoted by Md[P,,...,P,,]. This ring is then the homomorphic image of the polynomial ring Z[X,, . . , X,], where Z is the ring of integers. The purpose of this paper is to describe the kernel of that map and more generally to study the prime ideal structure of the ring M"[P,,. . ,PC]. We show that this ring is reduced and that the minimal prime ideals arise from the relations of rational dependency between the s-tuple of vertices v = (v,, . . . u,), one chosen from each P,,so that respectively they are extremal for P,over an open cone of directions in Rd. Modulo one of these minimal primes, the polytopes may be identified with their extremal points so that the quotient ring is the Minkowski ring of a finite number of points. It can be seen that this is a monomial algebra which under certain circumstances is an affine toric variety (see [l, 41). The analysis of the prime ideal structure is carried out by considering a valuation map E,, on M" to the ring Z[tr: r E R] for each u E R”. This is a ring homomorphism which sends a polytope P to t” where p is the extremal value of P in the direction of ~1. It corresponds to the end point valuation of the projection of the polytope P onto the line defined by the vector ~1. The map F, has a continuity property expressed by the fact that if e,,(f) # 0 for some u E E@, then F~,(f) # 0 on an open set in R”. It is this fact which allows one to show that prime ideals in M"[P,,...,P,$] are given by the collection of simple functions whose evaluation vanishes on an open cone of directions in subspaces L of R”. These are denoted by QL.v. We show (Proposition 3.6) how given a subspace T of the rational space Q”, one may associate to it an ideal in Z[X,, . . , X,] which is prime by virtue of the fact that it is defined as the collection of simple functions that vanish on the polytopes defined by the vertices given by T. We view this as an interesting application of Minkowski rings to a purely algebraic question. As another consequence, we establish (Corollary 3.7) the Krull dimension of the rings M,[P,, . ,P,]lQw,~,v as the codimension of the vector space of rational dependences between vertices that are open extremal. If the P, have rational vertices, then the dimension is realized as 1 + rank(M) where M is the (S X d) matrix which defines the vertices. The Minkowski ring of polytopes 241

1. Basic properties

If P is a polytope contained in R”, then one defines the support function of P [2] from KY’to R, by h,(u) = max{ (u, ~1): u E P}, where (., .) is the usual inner product. Since P is convex, hp(u) = max{ ( ui, u): u, a vertex of P} and hence hp(u) is continuous.

Definition 1.1. Let P = {P,, . . , P,} be an s-tuple of polytopes in Rd and let v= (u,, . . .) u,$) be an s-tuple of vertices, one from each polytope Pi, i = 1, . . . , s. We will say the pair {v, P} is extremal for u E R”, if for all i = 1,. . , s, hp,(u) = (u, v,). If W C Rd, we will say that {v, P} is extremal for W, if {v, P} is extremal for each u E IV. Finally, we say that {v, P} is open extremal, if it is extremal for some nonempty open set U CR”.

If u is a direction in Rd, i.e., a unit vector, then {v, P} is extremal for u if for each i=l,..., s, u, gives the maximal projection of the polytope Pi in the direction ~1. It is therefore clear that for each u E R”, there exists an s-tuple v which is extremal for u. Note that if v is extremal for U, then v is also extremal for (YU where (Yis a nonnegative scalar. Easy examples show that not every s-tuple of vertices v extremal for u need be open extremal.

Definition 1.2. A vector u E R” is said to be critical with respect to P if two distinct s-tuples of vertices v, , v2 are both extremal for ~1.

Observe that u is critical with respect to the s-tuple P if and only if it is critical with respect to a single polytope P in P. Therefore, u is critical when h,,(u) = (u, u,) = (u, u,) for two distinct vertices u, and u, of some PEP, i.e., when (u, u, - u,) = 0. Thus, by the continuity of h, the set of critical vectors is closed and its complement is dense.

Theorem 1.3. Let P be an s-tuple of polytopes in R”. Then the collection of open extremal vertex s-tuples is finite and for every vector u, there is an open extremal vertex s-tuple of P for u.

Proof. Clearly the collection of open extremal vertex s-tuples is finite since the number of polytopes is finite. Suppose {v, P} is extremal for u,,. It is easy to see that u,, is the limit of noncritical vectors {u,,} so that there is a unique {w, P} that is extremal for each u,,. By the continuity of h,, {w, P} is extremal for u,,. Therefore, it suffices to show that if u,, is not critical, then {v, P} is open extremal. But if u,, is not critical, then there exists an open set U containing u,, which contains no critical points and we need to show that v is extremal for each u E U. Suppose not; then without loss of generality we can assume that another 242 K. G. Fischer, J. Shapiro vertex s-tuple, which differs in the first position, is cxtremal for some vector in U. Let (w,,..., w,,,) be a list of all the vertices of P, which occur in an extremal s-tuple for some u E U. Let U, be the vectors in U which are mapped to zero by the function h,, - (., w!). Then these sets are closed and cover U and since they are finite in number, at least two of the sets have nonempty intersection. But if u is a vector in two of these sets, then u is critical for P, and hence critical for P, a contradiction. 0

We now consider the Minkowski ring of simple functions in the polytopes P in R”. If (Y = (a,, . , a,,) is an s-tuple of nonnegative integers we denote the product p;” . . . Pg5 by P”. Given a polynomial f in s variables, we will use f(P) to denote the simple function f(P,, . , P,). Such a function can be written as f(P) = c, ccyP” where the sum is taken over a finite number of distinct s-tuples (Y of nonnegative integers and where c, is a nonzero integer. Each ‘monomial’ P” is again a polytope which is the convex hull of the vectors

{N,U, +... + CY,u, : v an s-tuple of vertices from P,, . . , P, respectively} .

We will write cy *v to denote the vector (Y,u, + . . . + qu, and note that {v, P} is extremal for u if and only if {(Y *v, PC”} is extremal for ~1.

Definition 1.4. For each u E R”, define a function F,, : Md-+ Z[t’: r a real number], where for each f = c c, P” E M”, the image of f under F,, is given by

F,(f) = c C,, w

It is easy to see that F,, is a ring homomorphism. This follows by the definition of the product of two polytopes as their Minkowski sum. The following example in M’ will be used in the next proposition.

Example 1.5. Since polytopes in R’ are closed intervals and points, any f E M1 can be written in the form [a,, b,] +. . . + [ah, b,] ~ [c,, d,] - . . . - [c,,, d,,] (we allow a, = b, or c, = d,). If u is the unit vector 1 in R’, then

Ei,( f) rx [“I + . . . + p” - (4 - . _ p” = () means that k = n and for a suitable arrangement, b, = di for i = 1,. . , k. There- fore, if f is mapped to zero by F,,, then

for some integers p,. Conversely, if S is of this form, then clearly I,, = 0. The Minkowski ring of polytopes 243

Similarly one shows that I_ = 0 if and only if f can be written in the form C Pj(c,, ‘j]. F’ma 11y, note that the product of two elements in M’ of the form [a, b) and (c, d] is zero. This is seen directly by:

[a,b). cc,4 = ([a, 61 - Lb]). ([c, 4 - [cl> = [a + c, b + d] - [b + c, b + d] - [u + c, b + c] + [b + c]

=o.

If r : R”+ R” is a linear transformation, then there is an induced ring homomorphism 7~* : Md --+ M” given by P H 7~(P). If R” is a subspace of Rd and if u E R”, then it is not difficult to see that eI, equals E;) 0 n*, where QTis the projection of Rd onto R” and .sr’ IS the mapping on R” given in Definition 1.4. The important properties regarding the function E,, are expressed in the following proposition.

Proposition 1.6. Let u be a nonzero vector in Rd and let f E Md[P, , . . . , P,] where f(P) = c CUP”. If {v,P} 1s extremal for u, then eu( f) = c aut(“.“*‘). Additionally, f=Oiffe,(f)=OforalluER”.

Proof. The first part follows since {a *v, P”} is extremal for ~1. To prove the second statement we will first do the case d = 1. It suffices to show that ker(e,,) fl ker(e_,) = 0 if u is a unit vector. Let f be in this set. By Example 1.5, f can be written as a sum of intervals half open on the right and as a sum of intervals which are half open on the left. Hence for any b > 0, f. [0, b) = f. (0, b] = 0. Therefore, f. ([0] - [b]) = 0 and so, f = f. [b] for all b > 0. But clearly this is impossible unless f = 0. For d > 1, it can be deduced from [6, Theorem 31, that f = 0, when ‘ir*( f) = 0 for all projections 7~ onto one-dimensional subspaces of Rd. Since E,, is 7~* composed with F!‘), the proof now follows from the case d = 1. 0

Corollary 1.7. The ring M” is reduced.

Proof. It is clear from Proposition 1.6 that M” embeds in the product of the domains Z[t’: r a real number] indexed by the unit vectors in Rd. 0

2. Existence of algebraic relations

If 0 is the zero vector, we call E,,(f) = c, c, the Euler characteristic off and note that for any U, &()(f) = E,( f)l,=, If f has E u 1er characteristic zero, then for some positive integer k we may write

f = (M, + . . . + Mk) - (N, + . . . + Nk) , 244 K.G. Fischer, J. Shapiro where each M, and N, is a polytope in Rd. Define a continuous function

Hl. : R”+ R” via Hf(u) = (h,,(u), . . . , h,,(u), . . .) and let V, C [Wzkbe defined as

v, = {(x,, . . . >Xk,Yl,...~Yk): where(y,,...,y,)=a(x,,...,xk), some (TESk}.

Here, Sk is the set of all permutations on k elements. Observe that V, is a closed subset of [Wzk.The following explains the utility of introducing V,.

Proposition 2.1. Let f E Md be a simple function with Euler characteristic zero so thatf=(M,+..*+M,)-(N,+...+N,). Then: (i) Hf(u) E V, if and only if e,,(f) = 0. (ii) Let (u,, , vk, w,, . . . , wk) be an open extremal vertex 2k-tuple of vertices of the polytopes {M,, . . . , M,, N,, . . . , Nk} for the open set U. Then e,,(f) = 0 Vu E U if and only if there exists (T E Sk so that (w,, . . . , wk) = a(~,, . . . , uk).

Proof. To see (i) first note that

and hence E,,(f) = 0 if and only if (hN,(u), . . . , hNL(u)) is a permutation of (h,,(u), . . . , h,vk(u)). But the latter statement means that Hf(u) E V,. To show (ii) first assume that FU( f) = 0 Vu E I/. Then by (i)

H,(“)=((u,u,),..‘,(u,u,),(u,w,) ,..., (u,wk))Evk and since this function is linear in U, it follows that HJU) is an open subset of V,. For each CTE Sk, let V,, be the subset of V, defined as

Then y, is a closed set and V, = UcrESkV,, > Hf(U). It follows that there must exist some V* such that V,* n H,(U) has a nonempty interior. Hence there exists some open set W C R” so that

(T*((u, u, >, . . 9 ( u, uk:))= ((& w,), . . . I (u, w,)> for every u E W. Since this is true for all u in an open set W, it follows that (T*(u,, . . . ) Uk) = (w,, . . ) Wk). The Minkowski ring of polytopes 245

Conversely, if (T”(u,, . . . , vk) = (IV,, . . . , wk), then HJu) E V, for every u E U since (v, , . . . , vk, w, , . . , wk) is extremal for U. Therefore, q,(f) = 0 VU E U. 0 As a corollary we note the following interesting combinatorial criterion to determine whether a simple function with Euler characteristic zero is the zero function.

Corollary 2.2. Suppose that f = (M, + . . . + Mk) - (N, + * . . + Nk). Then f = 0 if and only if for every open extremal 2k-tuple of vertices (v,, . . . , vk, wl,. . . , wk) thereisaa~S,sothato(v ,,..., vk)=(w ,,..., wk).

Proof. The result follows easily from Theorem 1.3, Proposition 1.6 and (ii) of the above proposition. q

Theorem 2.3. Let {P, , . . . , P,} be a set of polytopes in Rd. There exists a nonzero polynomial f in s variables such that f(P) = 0 if and only if every open extremal s-tuple of vertices v of {P,, . . . , P,v} is linearly dependent over the integers.

Proof. Let {v, P} be extremal for the open set I/. Since the Euler characteristic of f(P) is zero we may write

f(P) = i P”’ - =$ PPJ ) 1=I I=1 where each (Y,and p, is an s-tuple of nonnegative integers and Vi, j oi # p,. Let v, (respectively vp) be the k-tuple whose nth coordinate is the vector 1y, *v (respectively p, *v). The 2k-tuple of vertices (v,, vp) is then an open extremal 2k-tuple of vertices of the 2k polytopes {Pal, P&: i, j = 1, . . . , k} for U. Since &Jf(P)) = 0 vu E u, i t f o I1ows by (ii) of Proposition 2.1 that the last k vertices vp are a permutation of the first k vertices v, . In particular, for each cq there exists a /3, such that q*(v ,,..., v,)=@,*(v ,,..., v,). Since (Y,#@,, this exhibits a nontrivial dependency of the vertices v, , . . . , v,. Conversely, suppose that v is open extremal. The hypothesis assumes that the vertices satisfy a linear relationship over the integers. Without loss of generality we may therefore write

0 = a,v, +. . . + a,v, - am+,v,+, - . . . - a,v, , where each of the a’s are positive integers. Let

f,(X) = x;l . . . x2 - x>m,l . . . x;\ ) where, if m = 0, then the first monomial in the expression is 1, while if m = s, 246 K. G. Fischer, J. Shapiro then the second monomial is 1. It follows that &(,(fV(P)) = 0 whenever v is extremal for U. Hence if f(X) = nvE,, f,(X), where A is the set of open extremal s-tuples of vertices of P, then by Theorem 1.3 and Proposition 1.6 one sees that f(P)=O. 0

The following corollaries are easy consequences of the theorem. The first is immediate and the second generalizes a result of Lawrence [6, Theorem 51.

Corollary 2.4. Let {P, , . . . , P,} be a set of polytopes in R”, where s > d. If each vertex of each Pi has rational coordinates, then the polytopes satisfy a nontrivial relationship. 0

Corollary 2.5. Suppose that P is a polytope with vertex set T = {u, , . . , u,,}. Let m be a positive integer and let P,, i = 1,2, . , m, be polytopes such that the vertices of each P, are contained in T. If the union of the vertices of the P,‘s equal T, then the set of polytopes {P, P,, . . , P,,} satisfies the nontrivial relationship n(P- P;)=O.

Proof. Let (u,,, u, , . . , u,,,) be an extremal m + l-tuple of vertices of {P, P, , . . , P,} for the nonempty open set U. The hypothesis assures that u(, = u, for some i > 0. 0

3. Continuity and prime ideals

The function E,, has a continuity property which allows us to exhibit prime ideals of Md[ P, , . , P,Y] in a natural way. For the remainder of the paper we will denote the ring Md[P,, . . , P,] by A. We need some more definitions that will be used throughout the section.

Definition 3.1. Let {v, P} be open extremal and let S be any nonempty subset of Rd. Then we define

WS,V={uES:visextremalforu} and

Q.s,V = {f E A: e,,(f) = 0 vu E WY,,) .

Observe that since F~, is multiplicative, Q.s,” is a radical ideal and if T C S, then tJkV C K., and Qs.” C Q,,,. Note also that QO,“, where 0 is the zero vector, is the ideal of all simple functions with Euler characteristic zero. It follows that if W,., f 8, then Qs,v C Q,,v and if W, y = $3,then Qs,V = A. We are most interested in sets S that are subspaces of Rd, in’which case 0 E W,s,,. Clearly W,., = S n W,~~,, and furthermore by Theorem 1.3 The Minkowski ring of polytopes 247 s = u w,.,, vt 1 where A is the set of open extremal s-tuples of vertices of the polytopes P.

We will show (Theorem 3.3) that the ideals QRLt,vare prime. Since A is finite and since, by Proposition 1.6, 0 = n vE,, QRC,,y,the minimal prime ideals of A are of the form QRd,Y. If I is the kernel of the natural map from Z[X,, . . . , X,] to A, then one sees that I is the intersection of prime ideals each the preimage of some QRd v. However, it is not true that all the prime ideals of this form are minimal primes (see Example 3.10).

Proposition 3.2. Suppose thut {v, P} is open extremal and f(X) E Z[X,, . , X,Y]. (i) If Ell,,(f(P)) # 0 f or some u,, E Rd, then there exists an open set U containing u,, so that q,( f(P)) f 0 Vu E U. (ii) Suppose that U is an open set in &for which E,,( f(P)) = 0 Vu E U. ZfS is any set so that U n W,,, # 0, then E,,( f(P)) = 0 Vu E W,,,

Proof. To prove (i) we may assume that f(P) has Euler characteristic zero for otherwise, cL,( f(P)) # 0 VU E R”. But E,,,&f(P)) Z 0 means, by (ii) of Proposition 2.5, that HJu,,) is not in V,. Since Hf is continuous and V, is closed, (i) follows. To prove (ii) note that since w.Y,, = S n W,,., it follows that if U n W.s,, is nonempty then so is U n W,,!,. Therefore, if we can prove (ii) under the hypothesis that S = R”, then the conclusion will hold for all S. Since v is an open extremal s-tuple and the set of noncritical points is dense, it follows that smce U fl WRC,,vis not empty it contains a nonempty open set. Because the Euler characteristic of f(P) is zero, we may write f(P) = Cfi=, P”’ - c F=, PO’ where q and pj are s-tuples of nonnegative integers. By Proposition 2.l(ii), the k-tuple (p, *v, . , Pk *v) is a permutation of the k-tuple ( “,“V,..., q *v) and therefore the k-tuple ((~1, pi *v)) is a permutation of ((u, CY,*v)) for all u E Rd. Thus by Proposition 2.1(i), &,,(f(P)) = 0 for all u for which v is extremal. Hence, F,,( f(P)) = 0 VU E IA’,,/.,. 0

Theorem 3.3. Let {v, P} be open extremal and let L be a real subspace of Rd. Then QL,v is a prime ideal of A and its preimage 9!L,v in Z[X,, . . . , X,] is generated by the difference of monomials.

Proof. We first do the case L = Rd. Suppose that f. g E QR,,,v, but g is not in Q,d,,. By definition there is some u E W,,,, so that q,(g) # 0 and therefore by (i) of Proposition 3.2 there is some open set U such that E.(,(g) f 0 Vu E U. Since 8, is multiplicative, e,,(f) = 0 Vu E U and since U n W,,,,, f 0, it follows by (ii) of Proposition 3.2, that F,,(f) = 0 VU E W,,,,. Hence, f E QWC~,Vproving that QRd v is a prime ideal. Now let L be a real subspace of Rd and let r : lRd+ L be the projection map. 248 K.G. Fischer, J. Shapiro

Then as noted in Section 2, rr induces a natural ring epimorphism r* : M”-+ M” where n = dim(L). Furthermore, 5-(v) is an extremal s-tuple of vertices of the polytopes {r(P,), . . , n(P,)}. Thus QL,p(Vj is a prime ideal of M”[n(P,), . . , ST(P,~)] by the previous paragraph. This clearly pulls back to the ideal QL,V of A and hence we have the first part of the result. For the second statement we can also assume that L = R. If F(X) is an arbitrary generator of 5?R,Vr then E~(F(P)) = 0 Vu E IV,,,. Using the same notation and proof as in Proposition 2.1, it follows that there exists a (T E S, such that e,,(M, - NC,(,)) = 0 for all u E W,,, and for all i = 1,. . , k. Hence, Mj - N,,(,, E Qw,v. Collecting these terms from all generators shows that s2,,, is generated by the differences of monomials. 0

The first part of the proof of the theorem shows that if f is a zero divisor in M”(P,, . . . , PJ], i.e., f. g = 0 with f,g # 0, then I,, must vanish on some W,,,. The converse of this is not true. That is, am, may vanish on some WRLj,,without being a zero divisor in M”[ P, . . . , P,]. We now want to examine the Krull dimension of A. In the process we give an interesting application of Minkowski rings which shows that certain ideals of Z[X,,.. . , X,] must be prime (see Proposition 3.6). We first need to establish some notation. Let {v, P} be open extremal and let M, be the rational subspace of R” generated by the vectors of v. If L is a real subspace of RBd,then we denote by L ’ the vectors in Rd which are perpendicular to L. Let Q’ denote the s-dimensional vector space over the rationals. Thus,

is a rational subspace of M,. Let 4 : Q’ + M, be the onto linear map which takes the ith basis vector of Q’ to the ith vector of v. Note that if L = R”, then $-‘(M, fl L’) is simply the kernel of 4. If T is a subspace of Q‘, let K, = Z” n T. These are the elements of T which have integral coordinates. If cy E 12‘. then there is a unique way of writing (Y as a = 4, - a,, > where LYI,and (Y,, have nonnegative integral coordinates and the jth coordinate, for any j, is nonzero in at most one of CY,,or cy,,. The polynomial X”J’ - X”jl will be denoted by jil (X).

Lemma 3.4. Let {v, P} be open extremal and L a real subspace of R“. If T= c#~‘(M, f~ L-), then for each ck E Z’, fn(P) E QL,, if and only if (Y E K,. Furthermore, QI_,v is generated by the elements { fU(P): (Y E K7}.

Proof. Write ff as a/, - LY,,.Then the 2-tuple (a,, *v, (Y,,*v) is an extremal vertex 2-tuple of { P”p, P”,l } whenever v is an extremal s-tuple of {P, , . . . , P,}. Further- more, for any u E W,,,, The Minkowski ring of polyropes 249

&Jf,) = t(ap*v.“) _ t(““*V.“)

The first part of the result now follows easily. To see the second statement observe that by Theorem 3.5 QL,” is generated by the difference of monomials Py - P”, where y and 6 are s-tuples of nonnegative integers. Clearly Py - Ps = P’fa(P) for some s-tuples (Y and /? of nonnegative integers. Since QL,V is prime and since P” has Euler characteristic 1, P, is not in Q L,v and therefore fa(P) must be in QL,“. The second statement follows. 0

Definition 3.5. Let T be an arbitrary subspace of Q’. Then the ideal of Z[X,, . . . , A’,] generated by elements of the form f,(X), where a E K,, will be denoted 2?(,,

Let {v, P} be open extremal and as before let 4 : Q’ + M,. It is clear from Lemma 3.4 that 22L,V, the pullback of QL,v, is generated by elements which are of the form f,(X) = X”p - X”?l, where a! = cy,, - (Y,!is in K,, for T = c#-‘(M, fl L ‘). Hence, 2!L,V = Qc7-, in this c ase. We give an application of Minkowski rings to show that every 9cr.j is of this form.

Proposition 3.6. Let T be a subspace of Q”, then 9(r) is a prime ideal of .qX,, . . . , X,]. Furthermore, if T and T’ are distinct subspaces of Q’, then S?(,, and 2Cro are distinct ideals.

Proof. Clearly, if T = $-‘(M, fl LA) for some L and v, then 2?(r) = .5?2,,,. In particular, 9c.1.j is a prime ideal by Theorem 3.3. We will prove the first statement by showing that .2(,, is always of this form and hence always prime.

Since R’ is infinite-dimensional over the rationals, it contains vectors U, , . . , u, such that the rational relations satisfied by these vectors are precisely given by T. Notice that u = (u,, . , u5) is an open extremal s-tuple of vertices of the polytopes {u, , . . . , us}. Hence, 2!(,, = 92,,, and therefore _G$r) is prime. To prove the second part of the statement, we know that f,(X) E 2?(,, if and only if a E K,. On the other hand, if T and T’ are distinct, then it is not difficult to see that K, and Kr, are distinct. The result follows. Cl

Corollary 3.7. Let A and v be as usual and let T = c#-‘(M, n L ‘) for some subspace L of Rd. Then the Krull dimension of A/Q,_, = 1 + s - dim(T).

Proof. Since Z[X,, . . , X,]/2+,, = A/Q,,,, the result follows from Proposition 3.6 and the fact that Z[X,, . . . , X,] has Krull dimension 1 + s. 0

Corollary 3.8. Let A = M”[P,, . . . , P,] and let A be a complete set of open extremal s-tuple of vertices of P. Then Krull dim(A) = 1 + max{dim(M,): v E A}. q 250 K.G. Fischer, J. Shapiro

We want to give two examples. The first indicates that the height of the minimal primes of I, the ideal of relations, can be mixed. The second example demon- strates some of the notation developed in the paper as well as answering some questions raised there.

Example 3.9. We consider a subring of M’. Let P, , P?, P, be the polytopes [2, 41, [3, 51 and [3, rr], respectively. Then the two extremal 3-tuples of vertices are v, = (4,5, r) and v2 = (2,3,3). Clearly, 9Rc,.V, is generated by X; - X;. In particular, $Rd,v, has height one. However, it can be seen that &,,Vz is generated by the elements {X2 - X,, XT - X:}. Hence, it has height two.

Example 3.10. In R’ let u,, u2, ui, w,, w2 and w3 be the vectors ( ;’ ), ( :‘), ( f ), ( T,’ ), (II) and (,“), respectively. Let P, be the convex hull of {u,, u7, w,, w3}, P, the convex hull of { u2, w?}, P, the convex hull of {u, , u2, w, , w2} and PA the convex hull of {u?, uj, w2, w,}. Then in the polytope ring we have P, + P2 - P, - P, = 0. We note that v = (u?, u2, u2, u3) is an open extremal vertex 4-tuple of {P, , P,, P,, P4} for the first quadrant and hence

We will describe the generators of the prime ideal %w~,Vof Z[X,, . . . , X,]. Let 4 : Q4+ M, be the map of Proposition 3.6. Thus, if we express v as the (2 x 4) matrix

ME 10 0 1 (1 1 1 1’1 then ker(+) = {a E Q’: M. a’ = 0} ( a’ is the transpose of a) which is generated by the vectors a=(l,O,O,-1) and b=(O,l, -l,O). In this casef,(X)=X, -X,, f!,(X) = X1 - X, and one can deduce that Scker(hjj = &&z,~= (X, - X,, X, - X,). If L is the line generated by ui in R’, then T = cf-‘(M, f? L’) = {a: (u\. AI). u’=O}={u: (2,1,1,2).u’=O}. Hence

L&) = i&, = (W, - x3, (X, - x4>, (X2 - x3>)

The vertex 4-tuple v* = (w?, w2, wz, wi) is open extremal for the fourth quadrant and one finds in this case that

is generated by (1, 0, 0, -l), (0, 1, 0,O) and (0,0, 1,O). It follows that %R1,v*= (X, -X,, X2 - 1, x, - 1). The Minkowski ring of polytopes 251

We summarize the significant points of this example: (i) Since 2&~,~ C 2&~,~*,the prime ideals associated to the extremal vertex 4-tuples need not yield minimal primes of the ideal of relations I. (ii) If L* is the line generated by u = ( A’,), one finds that

22L,v* = (X, - 1, x2 - 1, x3 - 1, x, - 1) = q,,“* .

Hence a distinct chain of subspaces of Rd need not yield distinct prime ideals of the Minkowski ring.

Acknowledgment

The authors would like to thank Philippe Loustaunau for many helpful conversations that we had during the development of this paper.

References

[l] V.I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (2) (1978) 97-154. [2] H.G. Eggleston. Convexity (Cambridge University Press. Cambridge, 1958). [3] H. Groemer, Minkowski addition and mixed volumes, Geom. Dedicata 6 (1977) 141-163. [4] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials and polytopes, Ann. of Math. (2) 96 (1972) 318-337. [5] J. Lawrence, Valuations and polarity, Discrete Comput. Geom. 3 (198X) 307-324. [6] J. Lawrence. Minkowski rings. Preprint. [7] P. McMullen, The polytope algebra, Adv. in Math. 78 (1989) 766130.